Subdivision Exterior Calculus for Geometry Processing

1. A computer-implemented method for processing a subdivision surface of an input geometry, the method comprising: receiving, via one or more processors, a control polygonal mesh associated with one or more subdivision levels; generating, via the one or more processors, a subdivision surface model from the control polygonal mesh based on one or more differential equations by: determining, at each subdivision level of the control polygonal mesh, one or more subdivision matrices of a respective k-form of each respective subdivision level, computing, from the subdivision matrices determined from each k-form, one or more matrices approximating differential operators, solving the one or more differential equations with the approximated differential operators computed from the subdivision matrices, and subdividing the control polygonal mesh using a solution of the solved one or more differential equations; and rendering, via the one or more processors, a subdivided control polygonal mesh with the subdivision surface model.

2. The method of claim 1 , wherein the differential equations are specified according to a geometric processing task.

3. The method of claim 2 , further comprising: applying the one or more differential operators as parameters to the geometric processing task.

4. The method of claim 2 , wherein the geometric processing task is one of at least a parameterization operation, a geodesic distance calculation operation, and a vector field design operation.

5. The method of claim 1 , wherein the subdivision is performed on an original version of the control polygonal mesh.

6. The method of claim 1 , wherein the subdivision matrices define refinable discrete forms on different levels of the control polygonal mesh.

7. A non-transitory computer-readable storage medium storing instructions, which, when executed on a processor, performs an operation for processing a subdivision surface of an input geometry, the operation comprising: receiving, via one or more processors, a control polygonal mesh associated with one or more subdivision levels; generating, via the one or more processors, a subdivision surface model from the control polygonal mesh based on one or more differential equations by: determining, at each subdivision level of the control polygonal mesh, one or more subdivision matrices of a respective k-form of each respective subdivision level, computing, from the subdivision matrices determined from each k-form, one or more matrices approximating differential operators, solving the one or more differential equations with the approximated differential operators computed from the subdivision matrices, and subdividing the control polygonal mesh using a solution of the solved one or more differential equations; and rendering, via the one or more processors, a subdivided control polygonal mesh with the subdivision surface model.

8. The computer-readable storage medium of claim 7 , wherein the differential equations are specified according to a geometric processing task.

9. The computer-readable storage medium of claim 8 , wherein the operation further comprises: applying the one or more differential operators as parameters to the geometric processing task.

10. The computer-readable storage medium of claim 7 , wherein the geometric processing task is one of at least a parameterization operation, a geodesic distance calculation operation, and a vector field design operation.

11. The computer-readable storage medium of claim 7 , wherein the subdivision is performed on an original version of the control polygonal mesh.

12. The computer-readable storage medium of claim 7 , wherein the subdivision matrices define refinable discrete forms on different levels of the control polygonal mesh.

13. A system, comprising: a processor; and a memory storing program code, which, when executed on the processor, performs an operation for processing a subdivision surface of an input geometry, the operation comprising: receiving, via the processor, a control polygonal mesh associated with one or more subdivision levels, generating, via the processor, a subdivision surface model from the control polygonal mesh based on one or more differential equations by: determining, at each subdivision level of the control polygonal mesh, one or more subdivision matrices of a respective k-form of each respective subdivision level, computing, from the subdivision matrices determined from each k-form, one or more matrices approximating differential operators, solving the one or more differential equations with the approximated differential operators computed from the subdivision matrices, and subdividing the control polygonal mesh using a solution of the solved one or more differential equations; and rendering, via the processor, a subdivided control polygonal mesh with the subdivision surface model.

14. The system of claim 13 , wherein the differential equations are specified according to a geometric processing task.

15. The system of claim 14 , wherein the operation further comprises: applying the one or more differential operators as parameters to the geometric processing task.

16. The system of claim 13 , wherein the subdivision is performed on an original version of the control polygonal mesh.

17. The system of claim 13 , wherein the subdivision matrices define refinable discrete forms on different levels of the control polygonal mesh.